I saw that in a real analysis proof, they used a proof by contradiction where the metric was a discrete metric. That is, distance is defined to be 1 if the points ARENT the same and 0 if the points are the same. I was trying to visualize how this would look for a unit circle. Does such a visualization exist? What is the correct way of viewing it in 2 dimensions? Thank you!

I think that you should not focus on the visualization. Strictly speaking the visualization does not change, the unit circle is always the same, the only difference is that you do not have -so to speak- a fine grained yardstick such that $d: X \times X \to \Re$, but only $d: X \times X \to \{ 0, 1 \}$.
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KolminJun 8 '14 at 10:11

The question doesn't really have anything to do with metrics; if I understand correctly you're asking if it's possible to have $n$ points with distance $1$ from each other. It's more geometry than anything.
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Najib IdrissiJun 8 '14 at 10:20

2 Answers
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In $\mathbb R^2$ simply imagine your space as consisting of two points at unit distance, or as three points, the vertices of an equilateral triangle. In $\mathbb R^3$, you visualize a discrete space with 4 points, the vertices of a perfect tetraeder. You can go to higher dimensions, for instance in $\mathbb R^n$ you can visualize a discrete metric space as being the vertices of a simplex. Going to $\mathbb R^\infty $ (or spaces of sequences such as $c$ or $\ell_p$), with a suitable metric, you can visualize a discrete metric space with infinitely many points.

Think of a collection of cities. Whenever you want to travel from one city to another, you have to use a strange and special train service. The train system is designed so that, no matter which two cities you travel between, the journey always takes one hour. Thinking of it another way ... no matter which city you're in, every other city is just one hour away.

In this scenario, the unit circle is the set of all cities, except the one you're currently in.

Thinking of "distance" as "travel time" is not unusual. If you ask how far it is from A to B, people will often say something like "it's about 20 minutes".